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A study on conserving invariants of the Vlasov equation in semi-Lagrangian computer simulations

Part of: PLA The Vlasov Equation

Published online by Cambridge University Press:  23 March 2017

L. Einkemmer Open the ORCID record for L. Einkemmer [Opens in a new window]
L. Einkemmer*
Affiliation:
University of Innsbruck, Innsbruck, Austria
*
†Email address for correspondence: lukas.einkemmer@uibk.ac.at
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Abstract

The semi-Lagrangian discontinuous Galerkin method, coupled with a splitting approach in time, has recently been introduced for the Vlasov–Poisson equation. Since these methods are conservative, local in space and able to limit numerical diffusion, they are considered a promising alternative to more traditional semi-Lagrangian schemes. In this paper we study the conservation of important physical invariants and the long-time behaviour of the semi-Lagrangian discontinuous Galerkin method. To that end we conduct a theoretical analysis and perform a number of numerical simulations. In particular, we find that the entropy is non-decreasing for the discontinuous Galerkin scheme, while unphysical oscillations in the entropy are observed for the traditional method based on cubic spline interpolation.

Keywords

plasma simulation
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 2 , April 2017 , 705830203
Copyright
© Cambridge University Press 2017 

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References

Brunetti, M., Califano, F. & Pegoraro, F. 2000 Asymptotic evolution of nonlinear Landau damping. Phys. Rev. E 62 (3), 4109.Google ScholarPubMed
Califano, F., Galeotti, L. & Mangeney, A. 2006 The Vlasov–Poisson model and the validity of a numerical approach. Phys. Plasmas 13 (8), 082102.CrossRefGoogle Scholar
Casas, F., Crouseilles, N., Faou, E. & Mehrenberger, M. 2017 High-order Hamiltonian splitting for Vlasov–Poisson equations. Numer. Math. 135 (3), 769–801.CrossRefGoogle Scholar
Cheng, C. & Knorr, G. 1976 The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22 (3), 330–351.CrossRefGoogle Scholar
Crouseilles, N., Faou, E. & Mehrenberger, M.2011a High order Runge–Kutta–Nyström splitting methods for the Vlasov–Poisson equation. http://hal.inria.fr/inria-00633934.Google Scholar
Crouseilles, N., Latu, G. & Sonnendrücker, E. 2009a A parallel Vlasov solver based on local cubic spline interpolation on patches. J. Comput. Phys. 228 (5), 1429–1446.CrossRefGoogle Scholar
Crouseilles, N., Mehrenberger, M. & Sonnendrücker, E. 2010 Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229 (6), 1927–1953.CrossRefGoogle Scholar
Crouseilles, N., Mehrenberger, M. & Vecil, F. 2011b Discontinuous Galerkin semi-Lagrangian method for Vlasov–Poisson. In ESAIM: Proceedings, vol. 32, pp. 211–230. EDP Sciences.Google Scholar
Crouseilles, N., Respaud, T. & Sonnendrücker, E. 2009b A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Commun. 180 (10), 1730–1745.CrossRefGoogle Scholar
Einkemmer, L. 2016a A mixed precision semi-Lagrangian algorithm and its performance on accelerators. In International Conference on High Performance Computing and Simulation (HPCS). IEEE.Google Scholar
Einkemmer, L. 2016b High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code. Comput. Phys. Commun. 202, 326–336.CrossRefGoogle Scholar
Einkemmer, L. & Ostermann, A. 2014a A strategy to suppress recurrence in grid-based Vlasov solvers. Eur. Phys. J. B 68 (7), 1–7.Google Scholar
Einkemmer, L. & Ostermann, A. 2014b Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov–Poisson equations. SIAM J. Numer. Anal. 52 (2), 757–778.CrossRefGoogle Scholar
Einkemmer, L. & Ostermann, A. 2014c Convergence analysis of Strang splitting for Vlasov-type equations. SIAM J. Numer. Anal. 52 (1), 140–155.CrossRefGoogle Scholar
Eliasson, B. 2002 Outflow boundary conditions for the Fourier transformed two-dimensional Vlasov equation. J. Comput. Phys. 181 (1), 98–125.CrossRefGoogle Scholar
Fijalkow, E. 1999 Numerical solution to the Vlasov equation: the 1D code. Comput. Phys. Commun. 116 (2), 329–335.CrossRefGoogle Scholar
Galeotti, L. & Califano, F. 2005 Asymptotic evolution of weakly collisional Vlasov–Poisson plasmas. Phys. Rev. Lett. 95, 015002.CrossRefGoogle ScholarPubMed
Galeotti, L., Califano, F. & Pegoraro, F. 2006 Echography of Vlasov codes. Phys. Lett. A 355 (4–5), 381–385.CrossRefGoogle Scholar
Hairer, E., Lubich, C. & Wanner, G. 2006 Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer.Google Scholar
Hou, Y. W., Ma, Z. W. & Yu, M. Y. 2011 The plasma wave echo revisited. Phys. Plasmas 18 (1), 012108.CrossRefGoogle Scholar
Klimas, A. J. & Farrell, W. M. 1994 A splitting algorithm for Vlasov simulation with filamentation filtration. J. Comput. Phys. 110 (1), 150–163.CrossRefGoogle Scholar
Manfredi, G. 1997 Long-time behavior of nonlinear Landau damping. Phys. Rev. Lett. 79 (15), 2815–2818.CrossRefGoogle Scholar
Mangeney, A., Califano, F., Cavazzoni, C. & Travnicek, P. 2002 A numerical scheme for the integration of the Vlasov–Maxwell system of equations. J. Comput. Phys. 179 (2), 495–538.CrossRefGoogle Scholar
Pollard, H. 1972 The convergence almost everywhere of Legendre series. Proc. Am. Math. Soc. 35 (2), 442–444.CrossRefGoogle Scholar
Qiu, J. M. & Shu, C. W. 2011 Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the Vlasov–Poisson system. J. Comput. Phys. 230 (23), 8386–8409.CrossRefGoogle Scholar
Rossmanith, J. A. & Seal, D. C. 2011 A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations. J. Comput. Phys. 230 (16), 6203–6232.CrossRefGoogle Scholar
Sonnendrücker, E.2011 Numerical methods for the Vlasov equation. http://icerm.brown.edu/html/programs/sp_f11/schedules/slides_workshop_1_tutorial/LectureNumericalVlasov.pdf.Google Scholar
Valentini, F., Veltri, P. & Mangeney, A. 2005 A numerical scheme for the integration of the Vlasov–Poisson system of equations, in the magnetized case. J. Comput. Phys. 210 (2), 730–751.CrossRefGoogle Scholar
Zerroukat, M., Wood, N. & Staniforth, A. 2005 A monotonic and positive-definite filter for a semi-Lagrangian inherently conserving and efficient (SLICE) scheme. Q. J. R. Meteorol. Soc. 131 (611), 2923–2936.CrossRefGoogle Scholar
Zerroukat, M., Wood, N. & Staniforth, A. 2006 The parabolic spline method (PSM) for conservative transport problems. Intl J. Numer. Meth. Fluids 51 (11), 1297–1318.CrossRefGoogle Scholar